Fractional-order memristive systems

This paper deals with the concept of (integer-order) memristive systems, which are generalized to non-integer order case using fractional calculus. We consider the memory effect of the fractional inductor (fractductor), fractional capacitor and fractional memristor. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operator is useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some simulations and experimental measurements as well.

[1]  A. Oustaloup.  La dérivation non entière , 1995 .

[2]  R. Williams,et al.  How We Found The Missing Memristor , 2008, IEEE Spectrum.

[3]  André Domhardt,et al.  Calculation of electrical circuits with fractional characteristics of construction elements , 2005 .

[4]  Karabi Biswas,et al.  Realization of a Constant Phase Element and Its Performance Study in a Differentiator Circuit , 2006, IEEE Transactions on Circuits and Systems II: Express Briefs.

[5]  L. Chua.  Memristor-The missing circuit element , 1971 .

[6]  K. B. Oldham,et al.  Analogue instrumentation for processing polarographic data , 1983 .

[7]  J. C. Wang.  Realizations of Generalized Warburg Impedance with RC Ladder Networks and Transmission Lines , 1987 .

[8]  B. Shenoi,et al.  Maximally flat lumped-element approximation to a fractional operator immittance function , 1970, IEEE Transactions on Circuit Theory.

[9]  H. W. Bode,et al.  Network analysis and feedback amplifier design , 1945 .

[10]  Frank Y. Wang.  Memristor for Introductory Physics , 2008 .

[11]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

[12]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[13]  YangQuan Chen,et al.  ANALOGUE FRACTIONAL-ORDER GENERALIZED MEMRISTIVE DEVICES , 2009 .

[14]  Alina Voda,et al.  Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function , 2007, Int. J. Appl. Math. Comput. Sci..

[15]  S. Roy.  On the Realization of a Constant-Argument Immittance or Fractional Operator , 1967, IEEE Transactions on Circuit Theory.

[16]  Tryphon T. Georgiou,et al.  The fractional integrator as a control design element , 2007, 2007 46th IEEE Conference on Decision and Control.

[17]  M. Nakagawa,et al.  Basic Characteristics of a Fractance Device , 1992 .

[18]  Alain Oustaloup,et al.  The CRONE Control of Resonant Plants: Application to a Flexible Transmission , 1995, Eur. J. Control.

[19]  I. Schäfer,et al.  Modelling of lossy coils using fractional derivatives , 2008 .

[20]  YangQuan Chen,et al.  Tuning and auto-tuning of fractional order controllers for industry applications , 2008 .

[21]  David M. Auslander,et al.  The Memristor: A New Bond Graph Element , 1972 .

[22]  I. Podlubny.  Fractional differential equations , 1998 .

[23]  Leon O. Chua,et al.  Putting Memory Into Circuit Elements: Memristors, Memcapacitors, and Meminductors MASSIMILIANO DI VENTRA , 2009 .

[24]  J. Tour,et al.  The fourth element , 2008 .

[25]  Xavier Moreau,et al.  The CRONE Suspension , 1996 .

[26]  George Oster,et al.  A note on memristors , 1974 .

[27]  L.O. Chua,et al.  Memristive devices and systems , 1976, Proceedings of the IEEE.

[28]  I. Podlubny.  Fractional-Order Systems and -Controllers , 1999 .

[29]  Leon O. Chua,et al.  Circuit Elements With Memory: Memristors, Memcapacitors, and Meminductors , 2009, Proceedings of the IEEE.

[30]  Y. Chen,et al.  A comparative introduction of four fractional order controllers , 2002, Proceedings of the 4th World Congress on Intelligent Control and Automation (Cat. No.02EX527).

[31]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[32]  J. Tour,et al.  Electronics: The fourth element , 2008, Nature.

[33]  S. Westerlund,et al.  Capacitor theory , 1994 .

[34]  B. T. Krishna,et al.  Active and Passive Realization of Fractance Device of Order 1/2 , 2008 .

[35]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[36]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[37]  Bharathwaj Muthuswamy,et al.  Memristor-Based Chaotic Circuits , 2009 .

[38]  Gregory S. Snider,et al.  Spike-timing-dependent learning in memristive nanodevices , 2008, 2008 IEEE International Symposium on Nanoscale Architectures.

[39]  Gary W. Bohannan,et al.  Analog Realization of a Fractional Control Element-Revisited , 2002 .

[40]  I. Podlubny,et al.  Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives , 2005, math-ph/0512028.

[41]  S. Westerlund.  Dead matter has memory , 1991 .

[42]  M. Ichise,et al.  An analog simulation of non-integer order transfer functions for analysis of electrode processes , 1971 .

[43]  Alexander Kushpel,et al.  Optimal Approximation on , 2000 .

[44]  Anissa Zergaïnoh-Mokraoui,et al.  State-space representation for fractional order controllers , 2000, Autom..